No, he doesn't. The Tokyo Revengers manga doesn't have Takemichi's death. Instead, he plays a crucial role in the unfolding events and continues to be an important part of the story.
Takemichi's time - leaping ability in the Tokyo Revengers real story is quite unique. It doesn't seem to follow a set pattern like in some other time - travel stories. As I said before, emotions play a big role. But also, it's not something he can fully control. Sometimes it happens unexpectedly. For example, when he's in a dangerous situation and his mind is focused on going back to change things, he is suddenly transported back. This ability is what drives the whole plot as he uses it to try and make a better future for his friends and Hinata by changing the past.
Peter Parker's reaction was one of determination. He was motivated to do more to prevent such tragedies in the future and was driven to be a stronger hero. His emotions were a mix of grief and a resolve to make a difference.
First, if the fanfiction is really good, you can recommend it to your friends who are also fans of Peter Parker. You can react by being critical too. If there are plot holes or if the character of Peter seems out of place, you can write a constructive review. Another way to react is by creating fanart inspired by the fanfiction. This shows your appreciation for the story in a creative way.